Table of polyhedron dihedral angles

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The dihedral angles for the edge-transitive polyhedra are:

Picture Name Schläfli
symbol
Vertex/Face
configuration
exact dihedral angle
(radians)
approximate
dihedral angle
(degrees)
Platonic solids (regular convex)
Tetrahedron.png Tetrahedron {3,3} (3.3.3) arccos(1/3) 70.53°
Hexahedron.png Hexahedron or Cube {4,3} (4.4.4) π/2 90°
Octahedron.png Octahedron {3,4} (3.3.3.3) π − arccos(1/3) 109.47°
Dodecahedron.png Dodecahedron {5,3} (5.5.5) π − arctan(2) 116.56°
Icosahedron.png Icosahedron {3,5} (3.3.3.3.3) π − arccos(√5/3) 138.19°
Kepler-Poinsot solids (regular nonconvex)
Small stellated dodecahedron.png Small stellated dodecahedron {5/2,5} (5/2.5/2.5/2.5/2.5/2) π − arctan(2) 116.56°
Great dodecahedron.png Great dodecahedron {5,5/2} (5.5.5.5.5)/2 arctan(2) 63.435°
Great stellated dodecahedron.png Great stellated dodecahedron {5/2,3} (5/2.5/2.5/2) arctan(2) 63.435°
Great icosahedron.png Great icosahedron {3,5/2} (3.3.3.3.3)/2 arcsin(2/3) 41.810°
Quasiregular polyhedra (Rectified regular)
Uniform polyhedron-33-t1.png Tetratetrahedron r{3,3} (3.3.3.3)  \pi - \arccos{\left( \frac{1}{3} \right)} 109.47°
Cuboctahedron.png Cuboctahedron r{3,4} (3.4.3.4)  \pi - \arccos{\left( \frac{1}{\sqrt{3}} \right)} 125.264°
Icosidodecahedron.png Icosidodecahedron r{3,5} (3.5.3.5)  \pi - \arccos{ \left( \sqrt{ \frac{ (5 + 2\sqrt 5)}{15} } \right) } 142.623°
Dodecadodecahedron.png Dodecadodecahedron r{5/2,5} (5.5/2.5.5/2) π arctan(2) 116.56°
Great icosidodecahedron.png Great icosidodecahedron r{5/2,3} (3.5/2.3.5/2)
Ditrigonal polyhedra
Small ditrigonal icosidodecahedron.png Small ditrigonal icosidodecahedron a{5,3} (3.5/2.3.5/2.3.5/2)
Ditrigonal dodecadodecahedron.png Ditrigonal dodecadodecahedron b{5,5/2} (5.5/3.5.5/3.5.5/3)
Great ditrigonal icosidodecahedron.png Great ditrigonal icosidodecahedron c{3,5/2} (3.5.3.5.3.5)/2
Hemipolyhedra
Tetrahemihexahedron.png Tetrahemihexahedron o{3,3} (3.4.3/2.4) 54.73°
Cubohemioctahedron.png Cubohemioctahedron o{3,4} (4.6.4/3.6) 54.73°
Octahemioctahedron.png Octahemioctahedron o{4,3} (3.6.3/2.6) 70.53°
Small dodecahemidodecahedron.png Small dodecahemidodecahedron o{3,5} (5.10.5/4.10) 26.063°
Small icosihemidodecahedron.png Small icosihemidodecahedron o{5,3} (3.10.3/2.10) 116.56°
Great dodecahemicosahedron.png Great dodecahemicosahedron o{5/2,5} (5.6.5/4.6)
Small dodecahemicosahedron.png Small dodecahemicosahedron o{5,5/2} (5/2.6.5/3.6)
Great icosihemidodecahedron.png Great icosihemidodecahedron o{5/2,3} (3.10/3.3/2.10/3)
Great dodecahemidodecahedron.png Great dodecahemidodecahedron o{3,5/2} (5/2.10/3.5/3.10/3)
Quasiregular dual solids
Hexahedron.png Rhombic hexahedron
(Dual of tetratetrahedron)
- V(3.3.3.3) π − π/2 90°
Rhombic dodecahedron.png Rhombic dodecahedron
(Dual of cuboctahedron)
- V(3.4.3.4) π − π/3 120°
Rhombic triacontahedron.png Rhombic triacontahedron
(Dual of icosidodecahedron)
- V(3.5.3.5) π − π/5 144°
DU36 medial rhombic triacontahedron.png Medial rhombic triacontahedron
(Dual of dodecadodecahedron)
- V(5.5/2.5.5/2) π π/3 120°
DU54 great rhombic triacontahedron.png Great rhombic triacontahedron
(Dual of great icosidodecahedron)
- V(3.5/2.3.5/2) π π/(5/2) 72°
Duals of the ditrigonal polyhedra
DU30 small triambic icosahedron.png Small triambic icosahedron
(Dual of small ditrigonal icosidodecahedron)
- V(3.5/2.3.5/2.3.5/2)
DU41 medial triambic icosahedron.png Medial triambic icosahedron
(Dual of ditrigonal dodecadodecahedron)
- V(5.5/3.5.5/3.5.5/3)
DU47 great triambic icosahedron.png Great triambic icosahedron
(Dual of great ditrigonal icosidodecahedron)
- V(3.5.3.5.3.5)/2
Duals of the hemipolyhedra
Tetrahemihexacron.png Tetrahemihexacron
(Dual of tetrahemihexahedron)
- V(3.4.3/2.4) π π/2 90°
Hexahemioctacron.png Hexahemioctacron
(Dual of cubohemioctahedron)
- V(4.6.4/3.6) π π/3 120°
Hexahemioctacron.png Octahemioctacron
(Dual of octahemioctahedron)
- V(3.6.3/2.6) π π/3 120°
Small dodecahemidodecacron.png Small dodecahemidodecacron
(Dual of small dodecahemidodecacron)
- V(5.10.5/4.10) π π/5 144°
Small dodecahemidodecacron.png Small icosihemidodecacron
(Dual of small icosihemidodecacron)
- V(3.10.3/2.10) π π/5 144°
Small dodecahemicosacron.png Great dodecahemicosacron
(Dual of great dodecahemicosahedron)
- V(5.6.5/4.6) π π/3 120°
Small dodecahemicosacron.png Small dodecahemicosacron
(Dual of small dodecahemicosahedron)
- V(5/2.6.5/3.6) π π/3 120°
Great dodecahemidodecacron.png Great icosihemidodecacron
(Dual of great icosihemidodecacron)
- V(3.10/3.3/2.10/3) π π/(5/2) 72°
Great dodecahemidodecacron.png Great dodecahemidodecacron
(Dual of great dodecahemidodecacron)
- V(5/2.10/3.5/3.10/3) π π/(5/2) 72°

References[edit]